# Chapter 35 Alternating Current Circuits

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1 hapter 35 Alternating urrent ircuits ac-ircuits Phasor Diagrams Resistors, apacitors and nductors in ac-ircuits R ac-ircuits ac-ircuit power. Resonance Transformers

2 ac ircuits Alternating currents and voltages An ac circuit consists of a combination of circuit elements and an ac-generator with a time dependent sinusoidal output where v cost i cost v, i are the instantaneous voltage and current, are the maximum (or peak) voltage and current ƒ, ω are the frequency and angular frequency, in Hz The electric power is supplied to the public is ac form for reasons that will become apparent soon The voltage and frequency vary by country: f Ex: / f are /60 Hz in the USA but 0-40/50 Hz in Europe Elements of circuit (resistors, capacitors and inductors) behave somewheat differently in ac-circuits than in dc-circuits We ll look at the specific behavior of each one of them and combinations

3 Phasor Diagrams oncept So, the ac-circuits are characterized by alternating voltages and currents A common way to represent and analyze periodically alternating quantities is by using phasors: vector-like representations with the length corresponding to the magnitude, and the angle ωt with respect to a selected direction being the angle phase The alternating physical quantity such as the emf delivered by the generator is the projection along the respective direction The resulting diagram is called a phasor diagram Phasors allow the summation of quantities alternating with different phases much like vector sums Quiz 1: heck out the adjacent current phasor. The magnitude of the instantaneous value of the current is a) ncreasing b) Decreasing c) onstant ω ε 0 ωt cost 0

4 rms urrent and oltage Note that, because the maximum current is reached alternately, the energy dissipated by an ac-current across a resistor is less than that dissipated by a dccurrent equal to max across the same R The root-mean square or rms current is the direct current that would dissipate the same amount of energy in a resistor as is actually dissipated by the ac-current rms Alternating voltages can also be discussed in terms of rms values rms The average power dissipated in resistor in an ac-circuit: ac-ammeters and voltmeters are designed to read rms values max max P rmsr Quiz : Since the 10 specified when one speaks about the voltage of an outlet represents the rms voltage, what is the maximum voltage of the respective source? a) 10 b) 170 c) 110 max max

5 Elements of ac-ircuits Resistors onsider a circuit consisting of an ac-source delivering a current i and a resistor R The current through and the voltage across the resistor vary such that they reach their maximum values at the same time: the current and the voltage are said to be in phase i, v R v R i cost v cost R R in phase i cost R i v R We conclude that the variable direction of the current has no effect on the behavior of the resistor T t The maximum current occurs for a small amount of time and the mean current is zero Ohm s aw for a resistor R in an ac-circuit takes the same form as in dc-circuits: v R and i-phasors are aligned R vr ir or rms R rms or R R ωt i v R

6 Elements of ac-ircuits apacitors onsider a capacitor in circuit with an ac-source Notice that, when the current decreases (twice per period) the capacitor voltage v increases and vice-versa Therefore, the voltage reaches its maximum ¼ of a period later than the current across the capacitor: we say that the voltage lags behind the current by π/ q q idt v v cos sin i cos t tdt t i, v v i cost i v v cos t ¼ cycle lag i cost ¼T T t We see that we can formulate Ohm s aw for a capacitor in an ac-circuit: X Here the resistance-like impeding effect of the capacitor is called the capacitive reactance and is given by However, Ohm s law won t work for instantaneous currents and voltages X 1 rms rms X v ωt i -phasor lags

7 Elements of ac-ircuits nductors onsider an inductor in circuit with an ac-source The current in the circuit is impeded by the back emf of the inductor As a result, the voltage across the inductor leads the current by π/: di v sint dt i, v v i cost i v v cos t ¼ cycle lead i cost ¼T T t Again, we can formulate Ohm s aw for an inductor in an ac-circuit: Here the resistance-like impeding effect of the inductor is called the inductive reactance and is given by However, Ohm s law won t work for instantaneous currents and voltages X X rms rms X v -phasor leads ωt i

8 Quiz 3: What happens with the maximum current when the frequency is decreased in a capacitive ac-circuit? a) ncreases b) Decreases c) Stays constant Problems: 1. Resistor in ac circuit: An ac voltage source is connected to a resistor R = 40 Ω. The source has an output: v sin 314 Hz t a) Find the rms voltage and current, and the angular frequency of the source of the source. c) How does the current vary with time? Represent its phasor.. nductor in ac circuit: An inductor is connected to a 0.0 Hz power supply that produces a 5.0 rms voltage. a) What is the angular frequency of the source? b) How does the current vary with time? Represent its phasor. 3. apacitor in ac circuit: When a 4.0 µf capacitor is connected to a generator whose rms output is 7, the rms current in the circuit is observed to be 0.5 A. a) What is the angular frequency of the source? b) How does the current vary with time? Represent its phasor.

9 The R Series ircuit Functionality The resistor, inductor, and capacitor can be combined in circuits where the three elements will compete in order to impose their respective behavior, resulting into a phase difference φ between voltage and current different from 0, π/ or π/ For instance, consider the R series circuit: R v cost φ = 0 v R v v φ = π/ omments: i cost The instantaneous voltage v R across the resistor is in phase with the current The instantaneous voltage v across the inductor leads the current by π/ The instantaneous voltage v across the capacitor lags the current by π/ φ = π/ φ total

10 The R Series ircuit Phasor diagram et s build the phasor diagram for an R series circuit contains the voltage phasors on the same diagram with a reference current The voltage across the resistor is along the current phasor since it is in phase with the current The voltage across the inductor is perpendicular anticlockwise on the current phasor since it leads the current by π/ The voltage across the capacitor is perpendicular clockwise on the current phasor since it lags behind the current by π/ Since the voltages are not in phase, to get the voltage across the combination of elements we can add the phasors like vectors: R where φ is the phase angle between the net voltage and current, since R has the same phase as the current 1 tan R = X R i cost f X > X such that >, the voltage leads the current by φ = X = Z φ R = R = X f X < X such that <, the voltage lags the current by φ R = R φ = Z = X

11 mpedance of series ac-ircuits The resistance of a circuit determining the ac current is given by its impedance Z To find the impedance and phase, we can use the phasor diagram and Ohm s law: R X X Z So, Ohm s aw can also be applied to the whole R circuit: Z Z rms rms where the impedance for R series can be rewritten This form for Ohm s law can be regarded as a generalized form applied to any ac-circuit even though the impedance may have a different form than the one of a series ac-combination Z X X R 1 tan R 1 0 π/ π/ π/ The results can be extrapolated to variants of the R series circuit: 0 π/

12 Power in ac-ircuits Single capacitors and inductors in ac-circuits are associated with no power losses: n a capacitor, energy is stored during one-half of a cycle and returned and returned to the circuit during the other half n an inductor, the ac-source does work against the back emf of the inductor and energy is stored in the inductor, but when the current begins to decrease in the circuit, the energy is returned to the circuit Therefore, the net average power delivered by the ac-generator is converted to internal energy in the resistor onsequently, the average power dissipated in a generic ac-circuit is av P vi cos t cost cos t cos 1 Pav cos P rms rms cos where cos φ is called the power factor of the circuit So, we see that phase shifts can be used to maximize power outputs, by making the power factor 1 1 φ cosφ ωt

13 Resonance in an ac-ircuit The resonance of an ac-circuit occurs at a certain frequency ω 0 where the current takes an extreme value for the respective arrangement of elements For instance, in a series R, the resonance is achieved when the current reaches a maximum which occurs when the impedance has a minimum value Based on the expression for impedance, we get Z R X X Z R R series min v lags i v in phase with i v leads i This occurs when X = X, such that the resonance frequency ω 0 for the R series circuit is given by: Theoretically, if R = 0, the current would be infinite at resonance, but real circuits always have some resistance Ex: a) Tuning a radio: a varying capacitor changes the resonance frequency of the tuning circuit in your radio to match the station to be received b) Metal Detector: The portal is an inductor, and the frequency is set to a condition with no metal present. When a metal is present, it changes the effective inductance, which changes the current. The change in current is detected and an alarm sounds

14 Problem: 4. Parallel R circuit: A resistor R, a capacitor and an inductor are connected in parallel across an ac source that provides a voltage v cost a) What are the phases of the currents through each element with respect to v? b) Use the respective current phasors to find out the current i through the source in terms of the currents through the elements and the phase angle with respect to v. c) Find the impedance of the circuit d) alculate the respective resonance angular frequency ω 0 and the angle phase φ 0 for this frequency. Then calculate the minimum current 0. R v cost

15 33.7 Transformers Properties An ac-transformer consists of two coils of wire wound around a core of soft iron The side connected to the input A voltage source is called the primary and has N 1 turns The other side, called the secondary, is connected to a resistor and has N turns The core is used to increase the magnetic flux and to provide a medium for the flux to pass from one coil to the other Properties: 1. The rate of change of the magnetic flux is the same through both coils, such that the potential differences across the primary and secondary are related by N N N 1 B B 1 1 t t N 1 N N 1 1 When N > N 1 > 1 the transformer is called a step up transformer N Symbol:. On the other hand, the energy must be conserved, so the power input and output must be the same, such that the primary and secondary currents are related by When N < N 1 < 1 the transformer is called a step down transformer N

16 Transformers omments and an important application omments: 1 N1 N Z Besides voltages and currents, the N N N N transformer also transforms impedance: While ideally the energy is conserved across a transformer and the only energy loss is via resistive dissipation, in reality a transformer also loses energy due to eddy currents in the iron core Application: When transmitting electric power over long distances, it is most economical to use high voltage and low current since this minimizes the R power losses n practice, voltage is stepped up to about at the generating station and stepped down to at the distribution station and finally to 10 at the customer s utility pole A modern solution to this loss of energy is to use superconductive cables that would reduce the resistance rather than current

17 Problem: 5. Transformer in an ac-circuit: A transformer is to be used to provide power for a computer disk drive that needs 5.8 (rms) instead of the 10 (rms) from the wall outlet. The number of turns in the primary is 400, and it delivers 500 ma (the secondary current) at an output voltage of 5.8 (rms). a) Should the transformer have more turns in the secondary compared to the primary, or fewer turns? b) Find the current in the primary. c) Find the number of turns in the secondary.

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